3. Filters -Resonant circuits
Resonant circuits will select relatively narrow bands
of frequencies and reject others.
Reactive networks are available that will freely pass
desired band of frequencies while almost
suppressing other bands of frequencies.
Such reactive networks are called filters.
.
4.
5. Ideal Filter
An ideal filter will pass all frequencies in a given
band without (attenuation) reduction in magnitude,
and totally suppress all other frequencies. Such an
ideal performance is not possible but can be
approached with complex design.
Filter circuits are widely used and vary in complexity
from relatively simple power supply filter of a.c.
operated radio receiver to complex filter sets used
to separate the various voice channels in carrier
frequency telephone circuits.
6. Application of Filter circuit
Whenever alternating currents occupying
different frequency bands are to be separated,
filter circuits have an application.
7. Neper - Decibel
In filter circuits the performance of the circuit is
expressed in terms of ratio of input –current to
output-current magnitude.
current
Output
current
Input
e
Performanc
If the ratios of voltage to current at input and
output of the network are equal then
2
1
2
1
V
V
I
I
(1)
8. If several networks are used in cascade as shown
if figure the overall performance will become
n
n
n
V
V
V
V
X
V
V
X
V
V
X
V
V 1
1
4
3
3
2
2
1
.....
(2)
9. Which may also me stated as
4
3
2
1
4
3
2
1 .
1
.
. A
A
A
A
A
A
A
A
Both the processes employing multiplication of
magnitudes. In general the process of addition or
subtraction may be carried out with greater ease than
the process of multiplication and division. It is
therefore of interest to note that
n
c
b
a
n
c
b
e
e
e
e
e
....
.....
is an application in which addition is substituted
for multiplication.
10. If the voltage ratios are defined as
etc
e
V
V
e
V
V
e
V
V c
b
a
;.......
;
;
4
3
3
2
2
1
Eq. (2) becomes
n
c
b
a
n
e
V
V
........
1
If the natural logarithm (ln) of both sides is taken,
then
(3)
n
d
c
b
a
V
V
..........
ln
2
1
11. Consequently if the ratio of each individual network is
given as “ n “ to an exponent, the logarithm of the
current or voltage ratios for all the networks in series
is very easily obtained as the simple sum of the
various exponents. It has become common, for this
reason, to define
N
e
I
I
V
V
2
1
2
1
(4)
under condition of equal impedance associated
with input and output circuits
12. The unit of “N” has been given the name nepers
and defined as
N
nepers
2
1
2
1
ln
ln
I
I
V
V
(5)
Two voltages, or two currents, differ by one neper
when one of them is “e” times as large as the other.
13. Obviously, ratios of input to output power may also
may also be expressed In this fashion. That is,
N
e
P
P 2
2
1
The number of nepers represents a convenient
measure of power loss or power gain of a network.
Loses or gains of successive networks then may be
introduced by addition or subtraction of their
appropriate N values.
14. “ bel “ - “ decibel “
The telephone industry proposed and has
popularized a similar unit based on logarithm to
the base 10, naming the unit “ bel “ for Alexander
Graham Bell
The “bel” is defined as the logarithm of a power
ratio,
number of bels =
2
1
P
P
log
It has been found that a unit one-tenth as large is
more convenient, and the smaller unit is called the
decibel, abbreviated “db” , defined as
15. 2
1
P
P
log
10
dB (6)
In case of equal impedance in input and output
circuits,
2
1
2
1
V
V
log
20
I
I
log
20
dB (7)
Equating the values for the power ratios,
10
10
2 dB
N
e
Taking logarithm on both sides
16. 8.686 N = dB
Or 1 neper = 8.686 dB
Is obtained as the relation between nepers and
decibel.
The ears hear sound intensities on a
logarithmically and not on a linear one.
20. Both T and networks can be considered as built of
unsymmetrical L half sections, connected together in
one fashion for T and oppositely for the network.
A series connection of several T or networks leads
to so-called “ladder networks” which are
indistinguishable one from the other except for the
end or terminating L half section as shown.
23. For a symmetrical network the image impedance
and are equal to each other and the
image impedance is then called characteristic
impedance or iterative impedance, .
That is , if a symmetrical T network is terminated in
, its input impedance will also be , or the
impedance transformation ration is unity.
The term iterative impedance is apparent if the
terminating impedance is considered as the
input impedance of a chain of similar networks in
which case is iterated at the input to each
network.
i
Z1 i
Z2
0
Z
0
Z
0
Z
0
Z
0
Z
26. Characteristic Impedance
for a symmetrical T section
2
1
2
1
2
1
2
1
0
4
1
(
4 Z
Z
Z
Z
Z
Z
Z
Z T
Characteristic impedance is that impedance, if
it terminates a symmetrical network, its input
impedance will also be
0
Z
0
Z
0
Z is fully decided by the network’s intrinsic
properties, such as physical dimensions and
electrical properties of network.
30. propagation constant
Under the assumption of equal input and output
impedances, which may be , for a symmetrical
network, the current ratio.
0
Z
The magnitude ratio does not express the complete
network performance , the phase angle between the
currents being needed as well. The use of
exponential can be extended to include the phasor
current ratio if it be defined that under the condition
of 0
Z
e
I
I
2
1
31.
j
Where is a complex number defined by
Hence
j
e
e
I
I
2
1
If
A
I
I
2
1
e
I
I
A
2
1
j
e
32. With Z0 termination, it is also true,
e
V
V
2
1
The term has been given the name
propagation constant
= attenuation constant, it determines the
magnitude ratio between input and output
quantities.
= It is the attenuation produced in passing
the network.
Units of attenuation is nepers
33. = phase constant. It determines the phase
angle between input and output quantities.
= the phase shift introduced by the network.
= The delay undergone by the signal as it
passes through the network.
= If phase shift occurs, it indicate the
propagation of signal through the network.
The unit of phase shift is radians.
34. If a number of sections all having a common Z
n
I
I
I
I
I
I
I
I 1
4
3
3
2
2
1
........
from which
n
e
e
e
e
........
3
2
1
and taking the natural logarithm,
the ratio of currents is
n
........
..........
4
3
2
1
Thus the overall propagation constant is equal to the
sum of the individual propagation constants.
35. Physical properties of symmetrical
networks
Use the definition of and the introduction of
as the ratio of current for a
termination leads to useful results
e
0
Z
36.
e
Z
Z
Z
Z
I
I
2
0
2
1
2
1 2
The T network in figure is considered
equivalent to any connected symmetrical
network terminated in a termination.
From the mesh equations the current ratio can
be shown as
0
Z
37. Where the characteristic impedance is given as
2
1
2
1
2
0
4
Z
Z
Z
Z
The propagation constant can be related to the
network parameters as follows:
2
1
2
2
1
2
1
2
1
2
2
1
2
1
)
2
(
2
1
ln
)
2
(
2
2
1
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
e
39. SC
OCZ
Z
Z
0
Thus the propagation constant and the
characteristic impedance Z0 can be evaluated using
measurable parameters
OC
SCandZ
Z
From these these two equations it can be
shown that
OC
SC
Z
Z
tanh
40. Filter fundamentals
Pass band – Stop band:
The propagation constant is
j
For = 0 or
There is no attenuation , only phase shift occurs.
It is pass band.
2
1 I
I
band
Stop
-
occurs;
n
attenuatio
,
I
ve;
when 2
1 I
41. is conveniently studied by use of the expression.
2
1
4
2
sinh
Z
Z
0
4
1
2
1
Z
Z
It can be proved from this, the pass band condition
is as follows:
where the two reactance are opposite type. The
phase shift in pass band is given by :
2
1
1
4
sin
2
Z
Z
42. Since may have number of combinations,
as L and C elements, or as parallel and series
combinations, a variety of types of performance
are possible.
2
1 Z
and
Z
43. Constant k- type low pass filter
(a) Low pass filter section; (b) reactance curves
demonstrating that (a) is a low pass section or has pass band
between Z1 = 0 and Z1 = - 4 Z2
44. Variation of and with frequency for the low
pass filter
46. Constant k high pass filter
(a) High pass filter; (b) reactance curves demonstrating that (a)
is a high pass filter or pass band between Z1 = 0 and Z1 = - 4Z2
47. m-derived T section
(a) Derivation of a low pass section having a
sharp cut-off section (b) reactance curves for (a)
